Pman - I have fixed several such holes in lucite and vulcanite. I use super glue. For my restore kit, I go to the dollar stores and get 3 packs of superglue, both regular and gel. I like those for the price and storage. If you open a new bottle of glue, even the best brands dry up after time. My way yo spend 33 cents and the rest is always fresh.
For your repair, the hold is small enough to use the one drop of the gel glue. Let it set overnight. the hole then sealed, I would add some drops of the thin glue to just above level. Let dry overnight. I use disposable emery boards to smooth it out, followed by sandpaper up to 1500 grit.
If a stem hole is to large to hold a drop of the gel glue, I fold wax paper small enough to back the hole. If hasn't ever stuck to the paper for me, but if it did, it is a food grade material. If I have concerns about the inside of the repair being to rough, I use micro-files to smooth it out if bristle pipe cleaners don't do the jobs.
Good luck!